均值方差准则下正则-奇异控制的均衡问题
Equilibrium for regular-singular control under mean-variance criterion: A unified approach via control laws
浏览论文内容
中文总结 AI 辅助
本文针对均值方差准则下的混合正则-奇异控制问题，在个人内博弈框架中提出新均衡概念，推导了均衡的数学刻画，并为再保险问题构造了显式耦合均衡解，退化情形下解可简化为独立单控制均衡组合。
中文摘要 AI 辅助
本文研究一类均值方差准则下的混合正则-奇异控制问题。我们在个人内博弈框架中寻找时间一致的均衡策略，并提出一种新的均衡概念。在该概念下，我们推导了验证定理和必要条件，对均衡进行了完整的数学刻画。为说明该理论，我们为再保险问题构造了显式耦合均衡解，其中正则控制依赖于奇异控制状态，且奇异控制的自由边界会随正则控制表达式的变化动态切换，产生非平凡耦合。在退化情形α₂=0时，该耦合解退化为两个独立单控制均衡的组合，且与参数趋于零时的极限一致。
英文摘要
This paper studies a class of mixed regular-singular control problems under mean-variance criteria, where the drift and diffusion are allowed to depend on both the regular control and the level of singular control. We seek time-consistent equilibrium strategies in an intrapersonal game setting and propose a novel equilibrium notion where both regular and singular controls are unified via control laws, or equivalently, mappings on the augmented state space. Under which, we derive a verification theorem and necessary conditions providing a full mathematical characterization of the equilibrium. We apply the theory to a reinsurance problem. The equilibrium solution turns out to be nontrivially coupled, where the regular control depends on the level of singular control, and the free boundary of the singular control switches dynamically in accordance with the variation of the regular control expression. The free boundary is characterized in a piecewise semi-explicit manner and is proved to be $C^{1}$ across the switching point. In the degenerate case $α_2=0$, the coupled solution reduces to the combination of two independent single-control equilibria and coincides with the limit as the parameter tends to zero.
